Symmetric chain complexes, twisted Blanchfield pairings, and knot concordance

TL;DR

This paper develops a combinatorial method to compute twisted Blanchfield pairings for 3-manifolds derived from knots, and applies it to detect non-slice knots among certain satellite knots using Casson-Gordon style representations.

Contribution

It introduces a new combinatorial algorithm for the twisted Blanchfield pairing and applies it to knot concordance, revealing subtle non-slice properties of specific satellite knots.

Findings

Computed twisted Blanchfield pairings for zero-surgery 3-manifolds from knots.

Identified non-slice knots among genus two satellite knots using Casson-Gordon representations.

Explicitly located infection curves in the second derived subgroup of the knot group.

Abstract

We give a formula for the duality structure of the 3-manifold obtained by doing zero-framed surgery along a knot in the 3-sphere, starting from a diagram of the knot. We then use this to give a combinatorial algorithm for computing the twisted Blanchfield pairing of such 3-manifolds. With the twisting defined by Casson-Gordon style representations, we use our computation of the twisted Blanchfield pairing to show that some subtle satellites of genus two ribbon knots yield non-slice knots. The construction is subtle in the sense that, once based, the infection curve lies in the second derived subgroup of the knot group, and that we identify these infection curves explicitly.